This paper establishes constraint and limit systems as the structural definition of admissibility within the Paton System. While prior work defines admissibility as the condition for system membership and identifies boundaries as the locations where admissibility is enforced, the structure that determines admissibility has not been isolated as a domain. This paper shows that constraints define the conditions under which system states are permitted, while limits define the thresholds beyond which continuation fails. Constraint systems therefore provide the governing structure of admissibility, and limit systems define its failure conditions. Together, they form the structural basis of system existence, persistence, and collapse. This provides a unified interpretation of possibility and failure across physical, biological, computational, and economic domains.